I think one of the best songs for Wolverine's "anthem/theme" is Metallica's "Until It Sleeps". I see it encompassing his inner war with his true nature and human nature.

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I'd have to say Senses Fail "Bite to break skin", it seems to portray those moments in his life where he does give in to his animal rage and goes berserk, you can really get that feeling towards the end of the song with the lines "Bite to break skin, darken the secret, my sullen face painted with passion, feelings will die, it's not your first, I AM YOUR SON, FOLLOW MY FOOTSTEPS"

I would say "Animal I have become" by Three days Grace, because like what sgoodlin82 said, it appears to describe how wolverine turns into an animal and can't control himself when he's in that state.